A novel photonic fractional-order temporal differentiator is proposed based on the inverse Raman scattering (IRS) in the side-coupled silicon microring resonator. By controlling the power of the pump light-wave, the intracavity loss is adjusted and the coupling state of the microring resonator can be changed, so the continuously tunable differentiation order is achieved. The influences of input pulse width on the differentiation order and the output deviation are discussed. Due to the narrow bandwidth of IRS in silicon, the intracavity loss can be adjusted on a specific resonance while keeping the adjacent resonances undisturbed. It can be expected that the proposed scheme has the potential to realize different differentiation orders simultaneously at different resonant wavelengths.